Approximate solution of generalized inhomogeneous radical quadratic functional equations in 2-Banach spaces

Abstract

In this paper, using Brzdȩk and Ciepliński's fixed point theorems in a 2-Banach space, we investigate approximate solution for the generalized inhomogeneous radical quadratic functional equation of the form f (ax 2 + by 2) = af (x) + bf (y) + D(x, y), where f is a mapping on the set of real numbers, a, b ∈ R + and D(x, y) is a given function. Some stability and hyperstability properties are presented.

On the stability of radical septic functional equations

“…( 9) is not stable for r = s = 7 with β = C = 1 for γ(x, y) = ( x r + y s ). Note that in (23) the conditions r, s < 7(β − log 2 C) do not hold, thus making the solution unstable. Remark 3.…”

“…Moreover, Cho et al [22] proved the generalized Hyers-Ulam stability for ( 5) and ( 6) both in quasi-β-Banach spaces and (β, p)-Banach spaces. Ding and Xu [23] provided a further generalization of (4) by the following inhomogeneous functional equation…”

On superstability of exponential functional equations

“…Hyers [2] obtained the first important result in this field. See [3][4][5][6][7][8][9][10] for more information on functional equations and applications. In 1979, Baker, Lawrence, and Zorzitto [11] proved the superstability of the exponential functional equation: Let X be a real vector space and f : X → R be an approximately exponential function, i.e., there exists a nonnegative number ε such that…”

Investigations on the Hyers–Ulam stability of generalized radical functional equations

“…x 2 + y 2 = f (x) + f (y), considered for functions with the set of reals R as the domain (see, e.g., [21]). For more information on the equations and examples of recent results we refer to [5,7,12,20]. We present a kind of very general approach to that subject.…”